Combinatorics and Representation Theory

组合学和表示论

• 批准号: 1068861
• 负责人: Brendon Rhoades
• 金额: $ 13.5万
• 依托单位: Massachusetts Institute of Technology
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2011
• 资助国家: 美国
• 起止时间: 2011-09-01 至 2011-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1068861&HistoricalAwards=false
• 关键词: Combinatorics; Representation; Theory

This proposal aims to investigate several aspects of the interplay between combinatorics and representation theory. First, the PI will investigate the enumerative cyclic sieving phenomenon introduced by Reiner, Stanton, and White as it applies to combinatorial actions on the cluster complexes of Fomin and Zelevinsky as well as actions on tableaux arising from the theory of crystals. The PI will use objects from representation theory such as Kazhdan-Lusztig bases and cluster algebras to investigate these combinatorial problems. Second, the PI will investigate connections between hyperplane arrangements and the Hilbert series of certain rings related to the diagonal coinvariant module.The enumeration of objects in a finite set is the fundamental problem of combinatorics and has many scientific applications outside of pure mathematics. Symmetry groups were introduced by the ancient Greeks and are the primary motivation for the study of representation theory. So, the study of enumeration up to symmetry is of paramount importance in combinatorial representation theory and algebraic combinatorics. This project uses representation theory to indirectly count the elements of finite sets, yielding a deeper and more elegant understanding of their enumeration.

该建议旨在研究组合学和表示理论之间相互作用的几个方面。 首先，PI将研究由Reiner，Stanton和White引入的列举循环筛分现象，以适用于对Fomin和Zelevinsky聚类复合物的组合作用，以及源自晶体理论引起的tableaux的作用。 PI将使用表示理论的对象，例如Kazhdan-Lusztig碱基和集群代数来研究这些组合问题。其次，PI将调查超平面布置与对角共线模块相关的某些环的希尔伯特系列之间的联系。有限集中对象的枚举是组合术的基本问题，并且在纯数学之外具有许多科学应用。 对称群是由古希腊人引入的，是研究理论研究的主要动机。 因此，对对称性的枚举研究在组合代表理论和代数组合学中至关重要。 该项目使用表示理论间接计算有限集的要素，从而对其枚举产生了更深入，更优雅的理解。